On positive solutions and the Omega limit set for a class of delay differential equations

This paper studies the positive solutions of a class of delay differential equations with two delays. These equations originate from the modeling of hematopoietic cell populations. We give a sufficient condition on the initial function for $t\leq 0$ such that the solution is positive for all time $t>0$. The condition is "optimal". We also discuss the long time behavior of these positive solutions through a dynamical system on the space of continuous functions. We give a characteristic description of the $\omega$ limit set of this dynamical system, which can provide informations about the long time behavior of positive solutions of the delay differential equation.Comment: 15 pages, 2 figure

Effects of memory on the shapes of simple outbreak trees

Genomic tools, including phylogenetic trees derived from sequence data, are increasingly used to understand outbreaks of infectious diseases. One challenge is to link phylogenetic trees to patterns of transmission. Particularly in bacteria that cause chronic infections, this inference is affected by variable infectious periods and infectivity over time. It is known that non-exponential infectious periods can have substantial effects on pathogens’ transmission dynamics. Here we ask how this non-Markovian nature of an outbreak process affects the branching trees describing that process, with particular focus on tree shapes. We simulate Crump-Mode-Jagers branching processes and compare different patterns of infectivity over time. We find that memory (non-Markovian-ness) in the process can have a pronounced effect on the shapes of the outbreak’s branching pattern. However, memory also has a pronounced effect on the sizes of the trees, even when the duration of the simulation is fixed. When the sizes of the trees are constrained to a constant value, memory in our processes has little direct effect on tree shapes, but can bias inference of the birth rate from trees. We compare simulated branching trees to phylogenetic trees from an outbreak of tuberculosis in Canada, and discuss the relevance of memory to this dataset

Spin-dependent Bohm trajectories for hydrogen eigenstates

The Bohm trajectories for several hydrogen atom eigenstates are determined, taking into account the additional momentum term that arises from the Pauli current. Unlike the original Bohmian result, the spin-dependent term yields nonstationary trajectories. The relationship between the trajectories and the standard visualizations of orbitals is discussed. The trajectories for a model problem that simulates a 1s-2p transition in hydrogen are also examined.Comment: 11 pages, 3 figure

ClassTR: Classifying Within-Host Heterogeneity Based on Tandem Repeats with Application to Mycobacterium tuberculosis Infections.

Genomic tools have revealed genetically diverse pathogens within some hosts. Within-host pathogen diversity, which we refer to as "complex infection", is increasingly recognized as a determinant of treatment outcome for infections like tuberculosis. Complex infection arises through two mechanisms: within-host mutation (which results in clonal heterogeneity) and reinfection (which results in mixed infections). Estimates of the frequency of within-host mutation and reinfection in populations are critical for understanding the natural history of disease. These estimates influence projections of disease trends and effects of interventions. The genotyping technique MLVA (multiple loci variable-number tandem repeats analysis) can identify complex infections, but the current method to distinguish clonal heterogeneity from mixed infections is based on a rather simple rule. Here we describe ClassTR, a method which leverages MLVA information from isolates collected in a population to distinguish mixed infections from clonal heterogeneity. We formulate the resolution of complex infections into their constituent strains as an optimization problem, and show its NP-completeness. We solve it efficiently by using mixed integer linear programming and graph decomposition. Once the complex infections are resolved into their constituent strains, ClassTR probabilistically classifies isolates as clonally heterogeneous or mixed by using a model of tandem repeat evolution. We first compare ClassTR with the standard rule-based classification on 100 simulated datasets. ClassTR outperforms the standard method, improving classification accuracy from 48% to 80%. We then apply ClassTR to a sample of 436 strains collected from tuberculosis patients in a South African community, of which 92 had complex infections. We find that ClassTR assigns an alternate classification to 18 of the 92 complex infections, suggesting important differences in practice. By explicitly modeling tandem repeat evolution, ClassTR helps to improve our understanding of the mechanisms driving within-host diversity of pathogens like Mycobacterium tuberculosis

Spin-dependent Bohm trajectories associated with an electronic transition in hydrogen

The Bohm causal theory of quantum mechanics with spin-dependence is used to determine electron trajectories when a hydrogen atom is subjected to (semi-classical) radiation. The transition between the 1s ground state and the 2p0 state is examined. It is found that transitions can be identified along Bohm trajectories. The trajectories lie on invariant hyperboloid surfaces of revolution in R^3. The energy along the trajectories is also discussed in relation to the hydrogen energy eigenvalues.Comment: 18 pages, 8 figure

Implications of Lorentz covariance for the guidance equation in two-slit quantum interference

It is known that Lorentz covariance fixes uniquely the current and the associated guidance law in the trajectory interpretation of quantum mechanics for spin particles. In the non-relativistic domain this implies a guidance law for the electron which differs by an additional spin-dependent term from that originally proposed by de Broglie and Bohm. In this paper we explore some of the implications of the modified guidance law. We bring out a property of mutual dependence in the particle coordinates that arises in product states, and show that the quantum potential has scalar and vector components which implies the particle is subject to a Lorentz-like force. The conditions for the classical limit and the limit of negligible spin are given, and the empirical sufficiency of the model is demonstrated. We then present a series of calculations of the trajectories based on two-dimensional Gaussian wave packets which illustrate how the additional spin-dependent term plays a significant role in structuring both the individual trajectories and the ensemble. The single packet corresponds to quantum inertial motion. The distinct features encountered when the wavefunction is a product or a superposition are explored, and the trajectories that model the two-slit experiment are given. The latter paths exhibit several new characteristics compared with the original de Broglie-Bohm ones, such as crossing of the axis of symmetry.Comment: 27 pages including 6 pages of figure

Comment on "On the subtleties of searching for dark matter with liquid xenon detectors"

In a recent manuscript (arXiv:1208.5046) Peter Sorensen claims that XENON100's upper limits on spin-independent WIMP-nucleon cross sections for WIMP masses below 10 GeV "may be understated by one order of magnitude or more". Having performed a similar, though more detailed analysis prior to the submission of our new result (arXiv:1207.5988), we do not confirm these findings. We point out the rationale for not considering the described effect in our final analysis and list several potential problems with his study.Comment: 3 pages, no figure